Problem 11 - A Graphical Approach to College Algebra

Question

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$\left(\frac{1}{2}\right)^{x}=5$$

Step-by-Step Solution

Answer

Exact form: \( x = \frac{\ln(5)}{\ln\left(\frac{1}{2}\right)} \); Approximation: \( x \approx -2.322 \).

Step 1: Take the Logarithm

To solve the equation \( \left(\frac{1}{2}\right)^{x}=5 \), take the natural logarithm of both sides. This gives us: \[ \ln\left(\left(\frac{1}{2}\right)^{x}\right) = \ln(5) \] Using the logarithmic identity \( \ln(a^b) = b\ln(a) \), we rewrite it as: \[ x \cdot \ln\left(\frac{1}{2}\right) = \ln(5) \]

Step 2: Solve for x

To isolate \( x \), divide both sides of the equation by \( \ln\left(\frac{1}{2}\right) \):\[ x = \frac{\ln(5)}{\ln\left(\frac{1}{2}\right)} \] This is the exact form of the solution.

Step 3: Calculate the Numerical Approximation

Use a calculator to find the numerical values of the logarithms:1. \( \ln(5) \approx 1.6094 \)2. \( \ln\left(\frac{1}{2}\right) \approx -0.6931 \)Plug these values into the expression for \( x \):\[ x \approx \frac{1.6094}{-0.6931} \approx -2.3219 \] Thus, the solution to the nearest thousandth is \( x \approx -2.322 \).

Key Concepts

Natural Logarithms

Natural logarithms are a key concept in solving exponential equations. They involve the mathematical constant \(e\), which is approximately 2.71828. A natural logarithm, denoted as \(\ln\), is the inverse operation of raising \(e\) to a power. This inverse relationship is what makes natural logarithms so useful when dealing with exponential equations. By using \(\ln\), we can transform exponential equations into linear forms, which are much easier to solve.

• In our exercise, taking the natural logarithm of both sides of the equation \( \left(\frac{1}{2}\right)^{x}=5 \) helps to bring down the exponent \(x\) using the property \( \ln(a^b) = b\ln(a) \).
• After applying \(\ln\), we transform the equation into a solvable linear form: \( x \cdot \ln\left(\frac{1}{2}\right) = \ln(5) \).

Natural logarithms are particularly handy in allowing us to solve for \(x\) in compound contexts, like the previous example, where the base of the exponent is a fraction. By understanding natural logarithms, students can break down complex equations into simpler arithmetic steps.

Exact Solutions

Exact solutions provide a precise expression of the answer without any approximation. These solutions are invaluable when exact values are necessary, be it for theoretical analysis or further calculations where precision is paramount. For exponential equations, finding an exact solution often involves using logarithmic identities, as shown in our problem.

• In the instance of \( \left(\frac{1}{2}\right)^{x}=5 \), after taking natural logarithms, the equation becomes \( x = \frac{\ln(5)}{\ln\left(\frac{1}{2}\right)} \).
• This is the exact solution because it expresses \(x\) in terms of known logarithms, providing a basis for precise value calculation or deeper mathematical exploration without resorting to estimation.

Exact solutions are particularly useful in mathematical proofs or when results need to be repeated with high fidelity across multiple problems or disciplines. By understanding how to express solutions exactly, learners enhance their ability to manipulate equations and understand the underlying relationships in mathematical expressions.

Numerical Approximation

Numerical approximation is the process of finding a close estimate to an exact mathematical expression. This is often necessary in practical scenarios where exact values might be cumbersome or impossible to measure precisely. Approximations offer manageable solutions to otherwise complex mathematical expressions.

• Once we have the exact solution \( x = \frac{\ln(5)}{\ln\left(\frac{1}{2}\right)} \), we use a calculator to find numerical approximations. This includes calculating the values of \( \ln(5) \) and \( \ln\left(\frac{1}{2}\right) \).
• Solving provides us the approximate value \( x \approx -2.322 \), rounded to the nearest thousandth.

Approximations are crucial in fields that require practical results without the impractical demand for infinite precision. They allow us to apply exact mathematical theory into real-world situations more efficiently and are a fundamental tool in numerical analysis and calculus.